THE DIMENSION OF A VARIETY

THE DIMENSION OF A VARIETY

Ewa Graczy´ nska Opole University of Technology

Institute of Mathematics Luboszycka 3, 45–036 Opole, Poland

e-mail: egracz@po.opole.pl http://www.egracz.po.opole.pl/

and

Dietmar Schweigert Technische Universit¨ at Kaiserslautern

Fachbereich Mathematik

Postfach 3049, 67653 Kaiserslautern, Germany

Abstract

Derived varieties were invented by P. Cohn in [4]. Derived varieties of a given type were invented by the authors in [10]. In the paper we deal with the derived variety V

of a given variety, by a fixed hypersub- stitution σ. We introduce the notion of the dimension of a variety as the cardinality κ of the set of all proper derived varieties of V included in V.

We examine dimensions of some varieties in the lattice of all varieties of a given type τ . Dimensions of varieties of lattices and all subvarieties of regular bands are determined.

Keywords: derived algebras, derived varieties, the dimension of a variety.

2000 Mathematics Subject Classification: Primary: 08B99, 08A40;

Secondary: 08B05, 08B15.

The results of this paper were presented during the Workshop AAA71

and CYA21 in B¸edlewo, Poland on February 11, 2006.

1. Notations

By τ we denote a fixed type τ : I → N , where I is an index set and N is the set of all natural numbers. In the paper we deal only with finite types, i.e., card(I) is finite. We use the definition of an n-ary term of type τ from [4, p. 6].

T (τ ) denotes the set of all term symbols of type τ . For a given variety V of type τ , two terms p and q of type τ are called equivalent (in V ) if the identity p ≈ q holds in V .

Definition 1.1. For a given type τ , F denotes the set of all fundamental operations F = {f

: i ∈ I} of type τ , i.e., τ (i) is the arity of the operation symbol f

, for i ∈ I. Let σ = (t

: i ∈ I) be a fixed choice of terms of type τ with τ (t

) = τ (f

), for every i ∈ I.

Recall from [10] (cf. [4, p. 13]), that for a given σ, the extension of σ to the map σ from the set T (τ ) to T (τ ), leaving all the variables unchanged and acting on composed terms as:

σ(f

(p

, . . . , p

)) = σ(f

)(σ(p

), . . . , σ(p

)) is called a hypersubstitution of type τ .

In the sequel, we shall use σ instead of σ for a hypersubstitution.

A hypersubstitution σ will be called trivial, if it is the identity mapping.

The set of all hypersubstitutions of type τ will be denoted by H(τ ).

For any algebra A = (A, Ω) = (A, (f

: i ∈ I)) ∈ V , of type τ , the algebra A

= (A, (t

: i ∈ I)) or shortly A

= (A, Ω

), for Ω

= (t

: i ∈ I) is called a derived algebra (of a given type τ ) of A, corresponding to σ, for any σ ∈ H(τ ) (cf. [10, 17]).

Definition 1.2. The variety generated by the class of all derived algebras A

, of algebras A ∈ V will be called the derived variety of V using σ and it will be denoted by V

, for any fixed σ ∈ H(τ ).

For a class K of algebras of a given type τ , D(K) denotes the class of all derived algebras of K for all possible choices of σ of type τ , i.e.:

D(K) = [

{K

: σ ∈ H(τ )}.

D is a class operator examined in [10] (cf. [16, 17]).

Let us note, that V

= HSP (σ(V )), for a given variety V and σ, where σ(V ) denotes the class of all derived algebras A

, for A ∈ V .

Recall from [12]:

Definition 1.3. For a given set Σ of identities of type τ , E(Σ) denotes the set of all consequences of Σ by the rules (1)–(5) of inferences of G. Birkhoff (cf. [1, 12]).

M od(Σ) denotes the variety of algebras determined by Σ.

A variety V is trivial if all algebras in V are trivial (i.e., one-element).

Trivial varieties will be denoted by T . A subclass W of a variety V which is also a variety is called subvariety of V .

V is a minimal (or equationally complete) variety if V is not trivial but the only subvariety of V , which is not equal to V is trivial.

We accept the following definition from [17]:

Definition 1.4. A derived variety V

is proper if V

is not equal to V , i.e., V

6= V .

Note, that V

may be not proper only for nontrivial σ.

Recall from [10]:

Definition 1.5. A variety V of type τ is solid if V contains all derived varieties V

for every choice of σ of type τ , i.e., D(V ) ⊆ V .

Definition 1.6. A variety V of type τ is fluid if the variety V contains no proper derived varieties V

for every choice of σ of type τ .

Fluid varieties appear naturally in many well known examples (cf. [11]).

Derived varieties are an important tool for describing the lattice of all subva- rieties of a given variety and therefore we expect some practical applications of the invented notion.

Note, that our definition of a fluid variety does not coincide with that of [17].

2. The Dimension

Definition 2.1. If V is a variety of type τ , then the dimension of V is the

cardinality κ of the set of all proper derived varieties V

of V included in V ,

for σ ∈ H(τ ). We write then that κ = dim(V ).

From the definitions above it follows that the trivial variety T of a given type is of dimension 0.

Theorem 2.1. Minimal varieties are of dimension 0. Fluid varieties are of dimension 0.

Later on we shall use the well-known conjugate property of [3] (cf. [9, p. 35]

and [11]) and quote as:

Theorem 2.2. Let A be an algebra and σ be a hypersubstitution of type τ . Then an identity p ≈ q of type τ is satisfied in the derived algebra A

if and only if the derived identity σ(p) ≈ σ(q) holds in A.

From the theorem above, it immediately follows:

Theorem 2.3. Let V be a variety and two hypersubstitutions σ

and σ

of type τ be given. If σ

(f

) ≈ σ

(f

), is an identity of V for every i ∈ I, then the derived varieties V

and V

are equal.

P roof. The proof follows by induction on the complexity of terms of type τ .

In the proof we use the relation ∼

on sets of hypersubstitutions which was introduced by J. P lonka in [15] and used in [3] to determine the notion of V - equivalent hypersubstitutions in order to simplify the procedure of checking whether an identity is satisfied in a variety V as a hyperidentity.

Recall from [13, p. 221], that an algebra A is locally finite iff every finitely generated subalgebra of A is finite. A class of algebras is locally finite iff each of its members is a locally finite algebra.

Theorem 2.4. Assume that a variety V (of a finite type) is locally finite.

Then V is of a finite dimension.

P roof. As V is locally finite, therefore every finitely generated free algebra

in V is finite and therefore for every n ∈ N there is only a finite number

of non-equivalent n-ary terms in V . Moreover, in V there are only finitely

many fundamental operations (by the assumption). Therefore in V there

is only a finite number of non-equivalent hypersubstitutions of type τ . In

cosequence there are only finitely many derived varieties of V and dim(V )

is finite.

3. Dimensions of varieties of lattices

We present some examples in lattice varieties as an answer to a problem posed by Brian Davey (La Trobe University, Australia) during the Conference on Universal Algebra and Lattice Theory (July 2005) at Szeged University, Szeged (Hungary).

Let L = (L, ∨, ∧) be a lattice. A variety L

derived from a variety L of lattices must not be a variety of lattices.

This follows from the fact, that there are only four non-equivalent binary terms in lattices, namely x, y, x ∨ y and x ∧ y. Given a hypersubstitution σ of type (2,2). If σ is trivial, then the derived algebra L

is L itself. If one takes σ generated by σ(x ∨ y) = x ∧ y and σ(x ∧ y) = x ∨ y, then L

is the dual lattice L

= (L, ∧, ∨). Otherwise the derived algebra L

is not a lattice at all, as some lattice axioms will be failed, unless L is trivial (i.e., one-element lattice).

We got immediately:

Example 3.1. Let V be a nontrivial variety of lattices. Then a derived variety V

is the dual variety of lattices V

or a variety which is not a variety of lattices.

Example 3.2. The variety L of all lattices in type (2,2) is fluid and not solid.

The variety L is fluid as it is selfdual, i.e., L = L

. It is not solid, as the commutativity laws for ∨ and ∧ are not satisfied as hyperidentities in lattices, for example.

Theorem 3.1. Every variety of lattices is fluid.

P roof. Let V be a variety of lattices. Consider the dual variety of V , i.e., the variety V

of all dual lattices of V . Then there are only two possibilities:

(i) V

⊆ V and consequently V = V

or

(ii) V and V

are incomparable in the lattice of all varieties of lattices.

Therefore we conclude, that either V is selfdual or V and V

are incompa-

rable. In consequence V is fluid and dim(V ) = 0.

4. Dimensions of subvarieties of regular bands

In this section we concentrate on the lattice of all subvarieties of regular bands, described in [6, 7] and [8].

Definition 4.1. Bands is the variety B of algebras of type (2), defined by:

associativity and idempotency (i.e., a band is an idempotent semigroup).

Following [5, p. 11], let us note, that the variety of bands has only six non- equivalent binary terms, therefore only six hypersubstitutions of type (2) in the variety of bands should be checked, namely: σ

− σ

defined as follows:

σ

(xy) = x, σ

(xy) = y, σ

(xy) = xy, σ

(x, y) = yx, σ

(xy) = xyx and σ

(xy) = yxy to be considered in order to determine all derived varieties of a given subvariety of regular bands.

Recall Proposition 3.1.5(i) from [4, p. 11, 77]:

Definition 4.2. A variety V of type (2) is called hyperassociative if the associativity law is satisfied in V as a hyperidentity.

Proposition 4.1. A variety of bands is hyperassociative if and only if it is contained in the variety RegB of regular bands.

The propositions above may be considered as a motivation of our interest in the lattice of all subvarieties of the variety of regular bands.

In order to determine the dimension of all subvarieties of RegB, we shall use the following two theorems of [11]:

Theorem 4.1. The variety of B of all bands constitutes a not fluid and not solid variety of type (2).

Theorem 4.2. A variety V of bands is fluid if and only if it is minimal.

Remark 4.1. Note, that a nontrivial variety V is of dimension 0 if and only if it is fluid.

Definition 4.3. An identity e of the form p ≈ q is called leftmost (rightmost) if and only if it has the same first (last) variable on each side. An identity which meets both of these conditions is called outermost.

First we express three technical lemmas:

Lemma 4.1. Let Σ be a set of identities of type τ which are leftmost

(or rightmost). Then the set E(Σ) consists only of leftmost (rightmost)

identities.

P roof. The proof follows from the observation that all rules of inference (1)–(5) preserve the property of being the leftmost (or rightmost) iden- tity. Therefore the closure of the set of left(right)most identities consists of left(right)most identities.

From [6, 7] and [8] it follows that every subvariety of the variety B of all bands is defined by one additional identity added to two axioms of bands (i.e., associativity and idempotency).

Lemma 4.2. Assume that V and W are varieties of regular bands, W is defined by a single identity p ≈ q, i.e., W = M od(p ≈ q) (in the variety of regular bands). Then:

V

⊆ W, for a given σ ∈ H(τ ),

if and only if the derived identity σ(p) ≈ σ(q) is satisfied in V , i.e., V |=

σ(p) ≈ σ(q).

P roof. V

= HSP (σ(V )) ⊆ W if and only if σ(V ) |= p ≈ q. By Theorem 2.2 we conclude that A

|= p ≈ q, for every algebra A

∈ σ(V ), if and only if A |= σ(p) ≈ σ(q), for every algebra A ∈ V , i.e., V |= σ(p) ≈ σ(q).

A simple generalization of the above lemma is the following:

Lemma 4.3. Assume that V and W are varieties of type τ , W is defined by a set Σ of identities of type τ , i.e., W = M od(Σ). Then:

V

⊆ W , for a given σ ∈ H(τ ),

if and only if the derived identity σ(p) ≈ σ(q) is satisfied in V , i.e.,

V |= σ(p) ≈ σ(q), for every identity p ≈ q ∈ Σ.

P roof. The proof is similar as that of Lemma 4.2, where p ≈ q is any

identity of the given axiomatic Σ.

The next three propositions show some regularities in the dimensions of all subvarieties of regular bands described in [7, p. 244] and [8]:

Definition 4.4. The variety RB in the variety B bands is defined by the identity: y ≈ yxy. It is called the variety of rectangular bands.

The fact that the variety RB is solid was proved in [5, p. 96].

We expressed the situation of theorems above on the diagram, which describes the bottom part of the lattice of all identities of bands, see [10] and [12, p. 244] Proposition 3.1.5 of [4]:

ps s

s s s

s

s s zxyz = zyxz

s s

s

zxyz = zxzyz

s s

V

V

V

V

xy = y yx = y

V

V

x = y SL RB RegB

N B

LZ RZ

T

Theorem 4.3. The variety RB is of dimension 2.

P roof. The variety of RB of rectangular bands have only two nontrivial

subvarieties, namely the variety LZ defined by the identities: yx ≈ y (called

the variety of left-zero semigroups) and the variety RZ defined by xy ≈ y

(called the variety of right-zero semigroups), respectively. Both of them are derived varieties of RB by the first and the second projection, respectively.

To prove that, let A

∈ (RB)

, for A ∈ RB. Then the identity yx ≈ y is satisfied in A

, as: σ

(yx) ≈ y ≈ y ≈ σ

(y) is satisfied in A and consequently in (RB)

. Similarly for σ

. We conclude that dim(RB) = 2.

Theorem 4.4. The varieties V

and V

of bands defined by the identities:

zxy ≈ zyx (1)

and

yxz ≈ xyz, respectively, (2)

are mutually derived by σ

. Moreover, dim(V

) = dim(V

) = 1.

P roof. Note, that the varieties V

and V

has only two proper nontrivial subvarieties, namely: the variety of left (right) zero-semigroups (respec- tively) and the variety SL of semilattices. The variety of semilattices, defined (in the variety of bands) by the commutativity law: xy ≈ yx is not a derived variety of V

, neither of V

. This follows from the fact, that if the variety SL of semilattices would be a derived variety of V

, then SL = (V

)

or SL = (V

)

. This is impossible, via Theorem 1.3, as the derived identity of xy ≈ yx by the hypersubstitions σ

(or σ

), i.e., σ

(xy) ≈ σ

(yx) (or σ

(xy) ≈ σ

(yx)) is of the form xyx ≈ yxy is neither leftmost nor rightmost and therefore, by Lemma 4.1 is not satisfied in V

as every identity satisfied in V

is leftmost. Similarly for V

. The proof follows from the fact that the only proper derived variety of V

included in V

is the variety LZ of left zero semigroups. Similarly, one can show that the only proper derived subvariety of V

by the second projection σ

is the variety RZ of right zero semigroups.

Finally we conclude that dim(V

) = dim(V

) = 1.

Definition 4.5. Varieties of dimension 1 will be called prefluid.

Theorem 4.5. The varieties V

and V

of bands defined by the identities:

yx ≈ yxy (3)

and

xy ≈ yxy, respectively, (4)

are mutually derived by σ

. Moreover, dim(V

) = dim(V

) = 1.

P roof. The variety (V

)

is the variety LZ of bands defined by yx ≈ y.

We obtain that: (V

)

is different from V

, therefore (V

)

is proper and (V

)

⊆ V

. Note, that the derived variety (V

)

is proper and is the variety RZ of right-zero semigroups but is not included in V

. Similarly as in the previous theorem we conclude, that the variety SL of semilattices is not a derived variety of V

, as all the identities of V

are left- most. In order to exclude that, the variety V

defined by the identity (1) zxy ≈ zyx is the derived variety of V

by σ

consider the derived identity σ

(zxy) ≈ σ

(zyx) of (1) by σ

, i.e., the identity zxyxz ≈ zyxyz. If this identity would be satisfied in V

, then the identity zxyz ≈ zyxz would be satisfied in V

, which is not true due to the results of [6]–[8]. Dually, the derived variety of V

by σ

is not the variety V

. Therefore we conclude, that dim(V

) = 1. Similarly one can prove that dim(V

) = 1.

Theorem 4.6. The varieties V

and V

of bands defined by the identities:

zxy ≈ zxzy (5)

and

yxz ≈ yzxz, respectively, (6)

are mutually derived by σ

. Moreover, dim(V

) = dim(V

) = 3.

P roof. The proof that (V

)

((V

)

) is the variety LZ(RZ) of left (right) zero semigroups follows from the proof of previous observations. Obviously:

σ

(V

) = V

, as the derived identity σ

(yxz) ≈ σ

(yzxz) of (6) by σ

gives rise to the identity (5) zxy ≈ zxzy and vice versa. Therefore V

and V

are mutually derived by σ

. We will show that the derived variety of V

by the hypersubstitution σ

is the variety V

, i.e., σ

(V

) = V

. To show this consider the derived identity of (3) by σ

, i.e., the identity σ

(yx) ≈ σ

(yxy).

This gives rise to the identity yxy ≈ yxyxy, which is obviously satisfied in V

.

Moreover, note that the derived identity of (1) by σ

, i.e., σ

(zxy) ≈ σ

(zyx)

gives rise to the identity zxyxz ≈ zyxyz, which can not be satisfied in V

,

as otherwise the identity zxyz ≈ zyxz would be satisfied in V

, which is

impossible by the results of [6]–[8] and it has been shown already in the

proof of Theorem 4.5. Similarly one can show, that the derived variety

of V

by σ

is the variety V

. We conclude that dim(V

) = 3. Similarly,

dim(V

) = 3.

Definition 4.6. The variety N B of normal bands is defined by the identity:

zxyz ≈ zyxz.

(7)

Theorem 4.7. dim(N B) = 4.

P roof. For solidity of the variety N B confront [5, p. 96]. It follows, that all derived varieties of the variety N B are included in the variety of N B. Similarly as before we show that (N B)

is the variety LZ of left-zero semigroups and (N B)

is the variety RZ of right-zero semigroups.

Both of them are proper subvarieties of NB. It is obvious that (N B)

= (N B)

= N B. We show only that (N B)

= V

, as the derived identity of (1) zxy ≈ zyx by σ

, i.e., σ

(zxy) ≈ σ

(zyx) gives rise to the identity zxyxz ≈ zyxyz satisfied in N B. In order to exclude that the variety LZ of left zero semigroups, defined by the identity yx ≈ y equals to (N B)

, notice that the derived identity of yx ≈ y by σ

is the identity yxy ≈ y, which is not satisfied in N B, as the variety of N B is defined by the set of regular identities (cf. [14]), which has only regular consequences.

Similarly (N B)

= V

, as the derived identity of (2) yxz ≈ xyz by σ

, i.e., σ

(yxz) ≈ σ

(xyz) gives rise to the identity zxzyzxz ≈ zyzxzyz satisfied in N B and we conclude that dim(N B) = 4.

Theorem 4.8. dim(RegB) = 4.

P roof. For solidity of the variety RegB confront [5, p. 96]. Two derived

subvarieties of RegB are LZ and RZ, by σ

and σ

, respectively. The

derived varieties of RegB via σ

and σ

are equal to RegB. We show that

(RegB)

= V

. To prove that, consider the derived identity of the identity

(3) yx ≈ yxy by σ

, i.e., the identity σ

(yx) ≈ σ

(yxy) which gives rise to

the identity yxy ≈ yxyxy which is satisfied in RegB. In order to show that

the derived variety of RegB by σ

is not equal to the variety V

, note that the

derived identity of (1) zxy ≈ zyx by σ

, i.e., the identity σ

(zxy) ≈ σ

(zyx)

gives rise to the identity zxyxz ≈ zyxyz which is not satisfied in RegB, as

it was shown in the proof of Theorem 4.6 that this identity is not satisfied

in V

, which is a subvariety of RegB. Similarly, one can show, that the

derived variety of RegB by σ

is the variety V

. This finishes the proof that

dim(RegB) = 4.

We expressed the situation of theorems above on the diagram:

ps s

s s s

s

s s

s s

s

s s

3 3

1

1 1 1

4

0 0

0 2 4

0

Acknowledgement

The first author expresses her thanks to the referee for his valuable comments.

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Received 23 February 2006

Revised 10 June 2006